Use the remainder theorem to:
find the remainder when f(x) is divided by x – c, and
determine whether x – c is a factor of f(x).
f(x) = x^4 - 8x^3 + 9x^2 - 69x + 24 ;
x - 8
Please show all of your work.
when f(x) is divided by x-8, the remainder will be
f(8).
so f(8) = 4096 - 4096 + 512 - 552 + 24 = -16
so x-8 is not a factor, or else f(8) would have been zero.
Thanks.... this help a lot...
To use the remainder theorem to find the remainder when f(x) is divided by x - c, we need to replace x with c in the polynomial f(x) and evaluate it. The value obtained will be the remainder.
Given:
f(x) = x^4 - 8x^3 + 9x^2 - 69x + 24
x - c = x - 8
Now, let's substitute x = c = 8 into f(x):
f(8) = (8)^4 - 8(8)^3 + 9(8)^2 - 69(8) + 24
= 4096 - 8(512) + 9(64) - 552 + 24
= 4096 - 4096 + 576 - 552 + 24
= 48
Therefore, the remainder when f(x) is divided by x - c (x - 8) is 48.
To determine whether x - c (x - 8) is a factor of f(x), we check if the remainder is zero. If the remainder is zero, it means that x - c is a factor of f(x). Otherwise, it is not a factor.
In this case, the remainder is not zero (remainder = 48), which means x - 8 is not a factor of f(x).